The simplest answer for why a 3rd place console "failed" is that its the 3rd place console. If the ultimate goal is to have the number 1 console then 3rd place constitutes a failure. There are so many ways to define failure that unless the console failed in every possible way (TG-16, 3DO, Jaguar) it's pointless to just say a console was a failure. The two key ways a company looks at a console, I think, are profit and if it can spawn a profitable successor. In both of those regards the GameCube was a success while the 2nd place Xbox was a failure (at least for the time being). On the other hand MS may have been more interested in using the Xbox in a negative manner, not to make them proft but to cause Sony to lose more money than they could afford thus giving MS an advantage in other fields. In that regard the Xbox has been a success (for the time being). Similarly with Sony, the PS3 in either 3rd or 2nd place won't make a profit and could doom the PS4, classic business definitions of failure, but if it gives blu-ray any boost early and Sony wins that struggle it could be considered a success. Although it's way to early and the sales are way too low to tell if that is happening at them moment.
Back to the main topic, the GameCube was a failure by other definitions of the term. It sold below expectations, it sold less than the N64, it failed to capture the hearts of the hardcore crowd which is why they consider the marginally higher selling financial sinkhole of the Xbox to be more successful (and they are they people in the video game media creating notions and trends), and it failed to have any effect on its rivals. Again though I think the main reason its considered a failure is that it didn't come in 1st.
